Exploring the electronic structure, mechanical behaviour, thermal and high-temperature thermoelectric response of CoZrSi and CoZrGe Heusler alloys

By using density functional theory, we have explored the structural, electro-mechanical, thermophysical and thermoelectric properties of CoZrSi and CoZrGe Heusler alloys. The ground state stability was determined by optimising the energy in various configurations like type I, II, and III. It was found that these alloys stabilized in the ferromagnetic phase in type I. We employed the Generalised Gradient Approximation and modified Becke-Johnson potentials to explore the electronic structure. The band structures of each of these Heusler alloys exhibit a half-metallic nature. Additionally, the computed second-order elastic parameters reveal their ductile nature of them. To understand the stability of the alloys at different pressures and temperatures, we investigated various thermodynamic parameters using the Quasi-Harmonic Debye model. We obtained the transport coefficients using the Boltzmann theory. Our findings indicate that these alloys can be used in spintronics and thermoelectric domains.


Structural properties
The half-Heusler alloys exhibit cubic MgAgAs phase within the space group F-43 m.Within this space group, two configurations can exist: α-phase and β-phase.In the α-phase, the atom can occupy Wycoff positions 4a (0,0,0), 4b (0.5,0.5,0.5) and 4c (0.25,0.25,0.25),whereas, in the β-phase, the atom can be found at positions 4a (0,0,0), 4b (0.75,0.75,0.75)and 4c (0.25,0.25,0.25),we used in the calculation only the α-phase, all position of α-phase is recorded in Table 1.To determine the more stable geometry of these alloys, we examined their atomic locations in three distinct α-phase configurations: type I (4b,4a,4c), type II(4b,4c,4a) and type III (4c,4a,4b).Figure 1 illustrates the crystal structure with different atomic configurations.For each of the three configurations, we determined several structural characteristics, including the equilibrium lattice constant (a in Å), the bulk modulus (B in GPa), and its pressure derivative (B' 0 ) at zero pressure as reported in Table 2.The geometric configuration of type I is found to be the most stable state for both of these alloys as delineated in Fig. 2 through three different types.The structure is optimized in three phases, i.e., spin-polarized (Ferromagnetic FM), nonspin-polarized (non-magnetic NM) and anti-spin-polarized (Antiferromagnetic AFM) as shown in Fig. 3.We calculated volume and total energy (E-V) results from the Birch-Murnaghan equation 39 as laid out in Fig. 2.

Electronic properties
Electronic characteristics determine the behaviour of the materials and bonds within the atoms.The electronic properties are examined in the optimized lattice parameter by calculating the material's total energy, electronic band profile and density of the state (DOS) of the material.We also used different approximations like GGA and mBJ to estimate the behaviour of the material.
Table 1.Wyckoff positions in various phases of half Heusler alloys.A: Band Profile Analysing the electronic and magnetic properties of a material is a way to assess its suitability for technological applications.We have estimated the energy band Profile using the two schemes GGA and mBJ, in which mBJ gives the correct estimated energy band profile.The electronic band structure is crucial for identifying the nature of a material based on its electronic properties.It is used to define a system's valence and conduction bands as well as some other electrical phenomena.The band profile of half Heusler alloys CoZrSi and CoZrGe are laid out in Figs. 4, 5, 6 and 7 with GGA and mBJ approximation.These Figures show that the Co-based half Heuslers alloys cross the fermi level in up-spin and down-spin channels in GGA approximation which exhibits a metallic nature and in mBJ approximation energy shifted towards the negative energy from low energy regions, and hence it generates a gap.Thus, it shows a gap in the spin-up channel which indicates the semiconductor nature.
Based on this outcome, it has been determined that these alloys exhibit a half-metallic character when analysed using the mBJ approximation method.We have introduced mBJ Because GGA underestimates the bandgap of the material.When we used the mBJ potential over GGA, it was found that both the alloys are half-metallic with an indirect bandgap of 1.05 eV and 0.92 eV for CoZrSi and CoZrGe at Г-X symmetric point in the mBJ scheme, as laid out in Figs. 3 and 4 and the calculated bandgap is reported in Table 6.

B: DOS (Density of states)
Furthermore, band structures of the alloys were also interpreted in terms of total DOS, partial DOS and atomic DOS as shown in Figs. 8, 9, 10 and 11. Figure 8 shows a comparison of the total DOS distribution for GGA and mBJ.The atomic density of states shows the distribution of individual atoms that can be plotted to further clarify the half-metallic character, which is shown in Fig. 9 for both the alloys for different atoms (Co, Zr, Si and Ge).       by dashed lines.As expected, cobalt and zirconium, with their predominant d-orbitals, are the major contributors to electron densities.Conversely, Germanium, belonging to the IV-A group, predominantly employs s-orbitals or p-orbitals as its main electron carriers.Furthermore, the distinct peaks observed in Cobalt and Zirconium correspond to d-orbitals, while in the Germanium element, the sharp peak is attributed to the s-orbital.Significantly, this s-orbital, despite its low energy, appears in an energy region far from the fermi energy level.The results observed in Figs. 10 and 11 affirm the anticipated outcomes, validating the expected contributions from specific orbitals in Cobalt, Zirconium, Silicon and Germanium elements.The plots illustrate that the d-states of transition metals show metallic behaviour in the spin-down channel.However, in the spin-up channel, these d-states undergo splitting at the Fermi level, leading to the semiconducting nature of alloys.The d-d hybridization is profoundly responsible for splitting the d-state and forming the band gap.According to this, alloys are half-metallic, exhibiting 100% spin polarization.

C: Magnetic property
The magnetic moment of a compound plays a major role in studying the magnetic properties of it.The spin magnetic moment of half Heusler alloys has been explained by the Galanakis model 50,51 , according to the variance between up-spin and down-spin states.Here, Z t is the total amount of outer shell electrons, and M T stands for the total magnetic moment 53 .This technique involves deducting 24 from the total valence electrons for full Heusler alloys and 18 from the total valence electrons for half Heusler alloys.As for the electronic configurations of the elements Co, Zr, Si, and Ge, they are as follows: In the CoZrSi and CoZrGe alloys, the total count of valence electrons (Zt) is 17.According to the SP rule (M T = Z t -18), the total magnetic moment for the CoZrSi and CoZrGe alloys is 1.00 μB.Both the GGA and mBJ approximations yield total magnetic moments of 1.00 μB, which aligns with our findings, consistent with the SP rule.Notably, the transition metal Co makes the most significant contribution to the overall magnetic moment.

D: Curie temperature
The band structure, DOS, and the numerical value of total magnetization support the alloys half-metallic nature.Another essential property of spin injectors from an application standpoint is T C .Hence, we have computed the T C for the CoZrSi and CoZrGe HH alloys using the mean field approximation (MFA) 54 .The MFA states that the difference between the energies of NM and FM states is related to T C .The equation written below can be used to compute the T C .
where, K B is the Boltzmann constant and E NM−FM = E NM − E FM is the energy difference between the NM and FM phases.Table 6 provides a list of the estimated values of T C .The computed values of T C are found to be greater than the ambient temperature, demonstrating the suitability of these materials for spintronic applications.

E: Cohesive energy
The factor cohesive energy (E Coh ), formation energy and mechanical stability are used to determine a material's stability theoretically.The E Coh of a material is defined as the energy needed to separate it into its constituent parts and indicates the binding strength of an alloy.It measures a material's stability 55 .Therefore, the cohesive energy is used to measure the intermolecular energy of substances.The formula below includes adding all the atomic energies and subtracting them from the alloy's total energy, which can be used to determine the E Coh values of the alloys under consideration.
where E Tot is the total energy of the substance under consideration, while E Co , E Zr and E Si/Ge are the energies of single atoms.The stability of the structure will increase with the cohesive energy value.The computed values of E Coh for CoZrSi and CoZrGe are recorded in Table 4.
Co : [Ar 3d 7 4s 2 , Zr : [Kr 4d 2 5s 2 , Si : [Ne 3s 2 3p 2 and Ge : [Ar 3d 10 4s 2 4p 2 .The enthalpy of formation energy (ΔE) of a compound is the energy change associated with the formation of one mole of the compound from its constituent elements in their standard states.It can be calculated using the following formula: where, E Total is the total energy of the compound, E A , E B , and E X are the energies of the constituent elements, a, b, and d are the stoichiometric coefficients of the constituent elements.If the enthalpy of formation energy is negative, then the compound is stable.This is because the formation of the compound is associated with a release of energy.The enthalpy of formation energies of CoZrSi and CoZrGe were calculated using this formula and found to be −3.78eV/atom and −4.64 eV/atom, respectively.These negative values indicate that CoHfSi and CoHfGe are both stable compounds.CoZrSi and CoZrGe are stable compounds with negative enthalpy of formation energies.This means that they are likely to form spontaneously when their constituent elements are mixed together.

Mechanical properties
Through the elastic constants, mechanical stability is revealed under different circumstances 40 .The Energy-strain relations are used to calculate the elastic constants by implementing modest amounts of strain to the equilibrium structure and examining how the total energy changes.This report attempts to estimate the elastic properties of CoZrSi and CoZrGe half Heusler alloys.As the materials possess cubic structures, it becomes necessary to compute the second-order derivative of Birch Murnaghan for elucidating other mechanical characteristics, thereby requiring C 11 , C 12 , and C 44 .The relevant stability criterion's criteria can be expressed mathematically as C 11 + 2C 12 > 0; C 11 , C 12 > 0; C 44 > 0 and C 11 > 0 are followed to define structural stability 41 .Both CoZrSi and CoZrGe alloys satisfy the criterion criteria for stability, proving that both cubic structure alloys are mechanically stable.All the elastic parameters are calculated in Table 5.
Another important factor, elastic anisotropy (A) plays a crucial role in determining the micro-cracks formation during the development process 42 .If A = 1, the crystal must be entirely isotropic; if A is less than unity then elastic anisotropy is predicted.As a result, it can be seen from the estimated values in Table 5 that both the CoZrSi and CoZrGe Heusler have a high anisotropy.Pugh's ratio (B/G) has a maximum value of 1.75 to categorise the plastic performance of a material 43 .B/G > 1.75 indicates ductility; otherwise, brittleness is indicated.These materials have a ductile nature because Pugh's ratios for these alloys are higher than the critical value.Frantsevich's ratio (G/B), a second criterion for stability utilised by the materials 44 , whose equivalent to mathematical value is less than 1.06 indicates that these alloys have limited resistance to shear deformation.
Poisson's ratio (σ) is another component that is used to distinguish the alloys for brittle or ductile nature 45 .The Poisson's ratio (σ) also indicates the nature of the bonding forces in the material.The alloy is still ductile if its value is more than 0.26; otherwise, it becomes brittle.The Poisson's ratio has a limit between 0.25 and 0.5 for the central force in solids.Our computed value shows that these alloys have central-type bonding forces.The brittle or ductile nature of the materials can be determined directly using Cauchy's pressure (C P ) 46 .A positive value of C p indicates a ductile nature, whereas a negative value indicates a brittle nature.Since the importance of (C P ) for CoZrSi and CoZrGe are determined to be 35.67 and 62.83, respectively, it can be predicted that both alloys are ductile as shown in Table 5.This Kleinman parameter (ξ) calculates internal strains for bond twisting relative to bond elongating and indicates the comparatively simple bond bending 47 .Reduced bond stretching and reduced bond bending in a structure show that ξ = 0 and ξ = 1, respectively.This parameter is 0.56 for CoZrSi and 0.68 for CoZrGe.Therefore, the mechanical properties that accompany elastic constant and their definition illustrate how resistant they are to a wide range of external forces, providing a suitable foundation for industrial applications.
The Debye temperature is calculated by using the average sound velocity ν m Velocity ν m is defined through transverse velocity ν t and longitudinal velocity ν l and it is given by the follow- ing equation: We calculated the value of θ D using the values of the essential parameters; it is equivalent to 486.30K for CoZrSi and 424.86 K for CoZrGe.We found that it exhibits a declining tendency as Silicon and Germanium increase in size.Using Fine's relation, we determined the melting temperature of these materials 48 using an elastic constant (C 11 ).
For CoZrSi and CoZrGe, the calculated values of melting temperature (T m ) are 1935.39± 300 K and 1688.04 ± 300 K, respectively.Since CoZrSi & CoZrGe have a high subscript value it is suggested that the material can maintain its ground state structure over a wide temperature range.
We have calculated the sound phase velocities for the longitudinal and transverse modes based on elastic constants.There are only three directions of elastic waves in cubic F-43 m symmetry, [111], [110], and [100].As the waves travel in further directions, they are quasi-transverse or longitudinal waves 49 .Further, we have calculated the average velocity or Debye velocity (V D ) by using the following relation: All the computed values of phase velocities are publicized in Table 6.

Thermodynamic properties
We estimated multiple thermodynamic potentials via the quasi-harmonic Debye model to describe the thermodynamic stability of the alloys.Hereby, Gibbs function is interpreted as 56 , where E (V) denotes energy, A Vib is the vibrational term, and Ɵ (V) is the Debye temperature.The specific heat capacity (C V ), thermal expansion coefficient (α) and Grüneisen parameter (γ) are included in the predicted thermodynamic potentials within the range of pressure and temperature are 0-20 GPa and 0-900 K respectively.www.nature.com/scientificreports/Specific heat (C V ) is a central parameter for assessing the intensity of phase transitions in lattice vibration that may take place in a material when temperature and pressure are applied.The Higher atomic or molecular mobility indicates the existence of high-temperature instability.The calculated value of C V for CoZrSi and CoZ-rGe are shown in Fig. 12 which helps us to understand the impact of the increase in atomic vibrations caused by heat absorption.The graph shows that for different values of pressure, C V tends to grow exponentially at low temperatures, whereas at high temperatures it attains a constant value.Every solid exhibiting the Dulong-petit law is defined by high-temperature variation 57 .
The Grüneisen parameter (γ) tells us about the phonon frequencies which are affected by crystal volume variation.It also reveals the amount of anharmonicity present in the source material.The fluctuation against temperature and pressure is shown in Fig. 13.At a lower temperature, these alloys exhibit a decreasing exponential pattern and at high temperature, they exhibit a constant value.The temperature effect outweighs the pressure effect, which exhibits nearly no change.The reported values of CoZrSi and CoZrGe at zero pressure (0 GPa) and 300 K are reported in Table 7.
The thermodynamic equation of state is predicted by the way the thermal expansion coefficient (α) varies, making it significant from both a theoretical and an experimental approach.Figure 14 depicts the graphical difference for both alloys, with a fast-growing tendency at lower temperatures and a tendency toward constant values at higher temperatures.This is the result of the enharmonic effects at lower temperatures being suppressed.For all alloys, the modulus of the thermal expansion coefficient (α) shows a steady decline with pressure from 0 to 20 GPa.The (0 GPa) and 300 K values of α for CoZrSi and CoZrGe were computed as recorded in Table 7.

Thermoelectric properties
Thermoelectric materials convert unused thermal energy into usable energy and can be used to create efficient and environmentally friendly energy sources 58 .This necessitates the material's ability to transmit charge and heat both effectively and optimally.Half Heuslers are one of the different kinds of alloys whose transport behaviour www.nature.com/scientificreports/ is being studied.The fluctuation of significant transport parameters is plotted from 50 to 900 K to demonstrate the material's thermoelectric performance.The various thermoelectric coefficients are distinguished as follows 59 : The figure of merit formula ZT 60 delivers a clear definition of the efficacy of the performance of the material.Electronic structure, particularly nearby to the fermi level, is the foundation for thermoelectric material's efficiency.Compared to the p-states and d-states the Co and Zr atoms stay close to fermi energy.Consequently, electrons that are thermally excited from these states will undergo a transition to a state referred to as the Seebeck coefficient (S). Figure 15 illustrates how the Seebeck coefficient varies with temperature for both spin configurations.In both HH alloys, S is positive and suggests that both spin channels include p-type charge carriers (holes).The entire Seebeck coefficient was additionally estimated using two-current models, which is represented as 61 .
Figure 16 illustrates the relationship between electrical conductivity (σ/τ) and temperature (T).This graph demonstrates that as T grows and the concentration of carriers also rises, σ/τ is increased.Electrical conductivity based on carrier concentration is given by the equation σ = neµ 62 , where e and n denote electronic charge and mobility, respectively.Figure 16 reveals that increases in the spin-up channel and decreases in the spin-down channel, respectively, corroborate the semi-metallic behaviour.The value of Seeback coefficient at 300 and 900 K are recorded in Table 8.
Solids have thermal conductivity (κ Total ) due to the two different components: (a) the movement of holes and electrons within the crystal that carries heat (κ e ) and (b) the movement of phonons (κ l ).These two components add up to the total thermal conductivity, which is given by κ Total = κ e + κ l .The carrier concentration is a significant factor that influences the total thermal conductivity in this case.The thermal conductivity increases the number Table 7.The evaluated value of thermodynamic parameters as Grüneisen parameter (γ), Specific heat (C V in J mol −1 K −1 ), and Thermal expansion (α in 10 -5 K −1 ) at zero pressure and room temperature.In this equation, A is constant with a value of 3.04 × 10 −8 , M is the average molar mass, θ D is the Debye tem- perature, V is the average volume, γ is the Gruneisen parameter, n is the number of atoms in the primitive unit  cell (where, n = 3 for half Heuslers and n = 4 for full/quaternary Heuslers) and T is the temperature.The calculated value of total thermal conductivity is recorded in Table 8 at 300 and 900 K for both the alloys.The constant A is described as; Near the Debye temperature, comparatively accurate results have been obtained.At low temperatures, it can anticipate a rough estimate of the necessary lattice conductivity.Figure 17 shows the fluctuations of thermal conductivity and total thermal conductivity, which reveals that the total lattice thermal conductivity of both forms was following a downward trend with a slight drop at low temperatures.Lattice thermal conductivity is prevalent at minimum temperatures which exhibits a fast decline with increasing temperature.The electrical component becomes increasingly prominent as the temperature rises, and this leads to an increase of κ tot .Lattice thermal conductivity generally decreases with the increase in temperature across the board for these materials.This phenomenon is caused by Umklapp processes, they show a 1/T dependence on the occupation of κ l during phonon scattering at elevated temperatures 63 .The electrical thermal conductivity, on the other hand, increases as the temperature rises.Consequently, these half Heuslers carry high to low thermal conductivity for CoZrSi and CoZrGe as the temperature changes.This is because of the high anharmonicity they exhibit.
An important thermoelectric factor is the power factor (PF), which is used to determine the amount of electrical energy produced.A Material is a strong candidate if it has a high-power factor.Figure 18 shows that over the selected temperature range, the PF of each of these alloys exhibits a rising trend.CoZrGe and CoZrSi both exhibit a rising PF trend from lesser values to higher values.The increase in electrical conductivity is the main source of the PF value's growing nature because the PF and electrical conductivity are related (PF = S2σ).These alloys have a high-power factor, indicating that they are composed for thermoelectric applications.The variation of power factor (PF) at 300 and 900 K is reported in Table 8 for both the alloys.The thermoelectric performance of a material is typically assessed using a metric called the figure of merit, denoted as ZT.The figure of merit (ZT) mathematically expressed as: ZT = S 2 σ T k .Figure 18 presents a graphical depiction of the figure of merit (ZT) as a function of temperature.At a temperature of 900 K, the observed ZT values for CoZrSi and CoZrGe alloys are 0.51 and 0.57, respectively.These substantial ZT values signify that both materials exhibit great potential for thermoelectric applications.

Conclusion
We have summarized a comprehensive report on the Half Heuslers alloys CoZrSi and CoZrGe by illustrating their numerous fundamental properties.According to the DFT calculations, this crystalline structure is extremely stable because of its total ground state energy and cohesive energy.The band structures, which show spin-polarised properties reveal a half-metallic behaviour which is additionally confirmed by the integral value of the magnetic moment of these alloys.Elastic properties, including Cauchy pressure (C P ) and Pugh's ratio (B/G), reveal the ductile behaviour of these alloys.Thermal properties reveal the stability of these alloys under different temperatures.Additionally, we evaluated the thermoelectric properties, which indicate the suitability of these alloys for thermoelectric applications, where they can convert waste thermal energy into effective electrical energy.Generally, these Heuslers can be useful for applications in solid-state electronic devices and renewable energy.

Figure 2 .
Figure 2. Energy as a function of volume in the 3 different phases for CoZrSi and CoZrGe.

Figure 3 .Figure 4 .
Figure 3. Energy as a function of volume in the FM, NM, and AFM phases for CoZrSi and CoZrGe.

Figure 5 .
Figure 5. Spin polarised band structure of CoZrSi by mBJ approximation in spin up (↑) And spin down(↓) channel.

Figure 6 .
Figure 6.Spin polarised band structure of CoZrGe alloy by GGA approximation in spin up (↑) and spin down(↓) channel.

Figure 12 .Figure 13 .
Figure 12.Variation of specific heat (C v ) with temperature for CoZrSi and CoZrGe alloys.

Figure 15 .Figure 16 .
Figure 15.Variation in Seeback coefficient with temperature in both spin for CoZrSi and CoZrGe alloys.

Figure 18 .
Figure 18.Variation in Power factor and figure of merits with temperature in both spin for CoZrSi and CoZrGe alloys.

Table 3
Calculated partial density of state (pDOS) in mBJ method for CoZrSi half Heusler alloy.

Table 3 .
Total, interstitial and individual magnetic moment (μB) and band gap (eV) of CoZrSi and CoZrGe half Heusler alloys.

Table 4 .
Computed value of Curie temperature T C (K), the energy difference between FM and NM states E NM−FM (eV per formula unit), cohesive energy (eV per atom) and Formation energy (eV per atom) for Both HH alloys.

Table 6 .
Estimated sound velocities (m/s) along different directions.